After a few years of dog ownership I purchased a Neptune steam cleaner. Not as portable as your model (wouldn't work on stairs), and it doesn't work as well as a rented cleaner. But it is always available, and gets out the mess. (In my dog's case, vomit is the excretion of choice.)
I generally do a pass with the soap on, then again with just water. That gets any residual soap out.
Pre-treating stained areas with "Nature's Miracle" or other enzyme cleaner seems to help a lot.